Assuming that the relationship or connection isn't just linguistic and that Finns, Estonians, Lapps, and Hungarians intermarried with other Northern European peoples, and that those Uralic peoples intermarried with Siberian tribes that eventually made their way to North America, why would anybody think there weren't genetic connections?
Also, a lot of Europeans are descended (in small part) from Tatars and Mongols, who might themselves in turn have been very distantly related to Asian wanderers who became American Indians. Tatars and Mongols belong to the Altaic language groups, and there's speculation that some American Indian groups also trace their ancestry back to the Altai.
Of course, all this is more complicated than I could possibly understand. Relationships between languages don't always mean close genetic relationships. But they do suggest possible connections.
Dale Drinnon and others have made very good arguments that the X haplotype crossed into North America via the northern Atlantic, not across the Bering Strait; and I posted some things to FR some months back about this.
In a nutshell: the current inhabitants of Scandinavia are principally Germanic, but from the Paleolithic era until relatively late in the Neolithic, there were other inhabitants, apparently of Uralic origin. They were maritime hunters, pursuing seals, whales, and such, and made it all the way to Labrador, Baffin Island and Ellesmere Island in their sea-going canoes. They left genetic (and linguistic) traces in a wide arc stretching from Finland to Scotland (and they may have been the ancestors of the Picts!) to eastern Canada. In one of Drinnon’s blogs, he produced a long list of words that were the same (or very nearly so) in both Finnish and Ojibway!